Inverse temperature

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It is often convenient to define a dimensionless inverse temperature, :

This notation likely comes from its origin as a Lagrangian multiplier, for which Greek letters are customarily written.

Indeed, it shown in Ref. 1 that this is the way it enters. The task is to maximize number of ways $N$ particles may be asigned to $K$ space-momentum cells, such that one has a set of occupation numbers . Introducing the partition function:

one could maximize its logarithm (a monotonous function):

where Stirling's approximation for large numbers has been used. The maximization must be performed subject to the constraint:

An additional constraint, which applies only to dilute gases, is:

,

where is the total energy and is the energy of cell .

The method of Lagrange multipliers entail finding the extremum of the function

,

where the two Lagrange multipliers enforce the two conditions and permit the treatment of the occupations as independent variables. The minimization leads to

Failed to parse (syntax error): {\displaystyle n_i=Ce^{-\beta e_i), }

and an application to the case of an ideal gas reveals the connection with the temperature,


References

  1. Kerson Huang, "Statistical Physics" John Wiley and Sons, second edition, pp. 79-85 (1987)